Adaptive Gauss-Hermite Quadrature Research Articles

Variability in Causal Effects and Noncompliance in a Multisite Trial: A Bivariate Hierarchical Generalized Random Coefficients Model for a Binary Outcome.

Within each of 170 physicians, patients were randomized to access e-assist, an online program that aimed to increase colorectal cancer screening (CRCS), or control. Compliance was partial: of the experimental patients accessed e-assist while no controls were provided the access. Of interest are the average causal effect of assignment to treatment and the complier average causal effect as well as the variation of these causal effects across physicians. Each physician generates probabilities of screening for experimental compliers (experimental patients who accessed e-assist), control compliers (controls who would have accessed e-assist had they been assigned to e-assist), and never takers (patients who would have avoided e-assist no matter what). Estimating physician-specific probabilities jointly over physicians poses novel challenges. We address these challenges by maximum likelihood, factoring a "complete-data likelihood" uniquely into the conditional distribution of screening and partially observed compliance given random effects and the distribution of random effects. We marginalize this likelihood using adaptive Gauss-Hermite quadrature. The approach is doubly iterative in that the conditional distribution defies analytic evaluation. Because the small sample size per physician constrains estimability of multiple random effects, we reduce their dimensionality using a shared random effects model having a factor analytic structure. We assess estimators and recommend sample sizes to produce reasonably accurate and precise estimates by simulation, and analyze data from a trial of a CRCS intervention.

An evaluation of computational methods for aggregate data meta-analyses of diagnostic test accuracy studies

BackgroundA Generalized Linear Mixed Model (GLMM) is recommended to meta-analyze diagnostic test accuracy studies (DTAs) based on aggregate or individual participant data. Since a GLMM does not have a closed-form likelihood function or parameter solutions, computational methods are conventionally used to approximate the likelihoods and obtain parameter estimates. The most commonly used computational methods are the Iteratively Reweighted Least Squares (IRLS), the Laplace approximation (LA), and the Adaptive Gauss-Hermite quadrature (AGHQ). Despite being widely used, it has not been clear how these computational methods compare and perform in the context of an aggregate data meta-analysis (ADMA) of DTAs.MethodsWe compared and evaluated the performance of three commonly used computational methods for GLMM - the IRLS, the LA, and the AGHQ, via a comprehensive simulation study and real-life data examples, in the context of an ADMA of DTAs. By varying several parameters in our simulations, we assessed the performance of the three methods in terms of bias, root mean squared error, confidence interval (CI) width, coverage of the 95% CI, convergence rate, and computational speed.ResultsFor most of the scenarios, especially when the meta-analytic data were not sparse (i.e., there were no or negligible studies with perfect diagnosis), the three computational methods were comparable for the estimation of sensitivity and specificity. However, the LA had the largest bias and root mean squared error for pooled sensitivity and specificity when the meta-analytic data were sparse. Moreover, the AGHQ took a longer computational time to converge relative to the other two methods, although it had the best convergence rate.ConclusionsWe recommend practitioners and researchers carefully choose an appropriate computational algorithm when fitting a GLMM to an ADMA of DTAs. We do not recommend the LA for sparse meta-analytic data sets. However, either the AGHQ or the IRLS can be used regardless of the characteristics of the meta-analytic data.

The global effects of digestive system cancers worldwide on the COVID-19 pandemic by negative binomial (mixed) regression models

In this study, the main goal is to determine statistically significant relationships between the biggest epidemic problem of last years as the COVID-19 pandemic and digestive system cancers belonging to 168 countries worldwide using generalized linear mixed model (GLMM) and its special case as generalized linear model (GLM) approaches, and to obtain global inferences that will shed light on this pandemic. For this goal, the response variable is “total cases of the COVID-19 pandemic per 100,000 people” until January 14, 2022. The explanatory variables are total number of people suffering from colon and rectum, stomach, lip and oral cavity, esophageal, and nasopharynx cancers belonging to 2019, respectively. In this study, the negative binomial (NB) regression model in GLM using iteratively reweighted least squares (IRLS) algorithm and the NB mixed regression model in GLMM with “countries” taken as “random effects” using Adaptive Gauss-Hermite Quadrature (AGHQ) approximation method at 1, 2, 10, and 20 quadrature points are used for modelling the COVID-19 pandemic and digestive systems cancer data. In this study, by using information criteria, the NB mixed regression model in GLMM under log-link function with 168 countries taken as random effects using AGHQ method at 20 quadrature point is detected as the most appropriate model for the global COVID-19 pandemic and digestive system cancers data.

Maximum Likelihood Estimation of Hierarchical Linear Models from Incomplete Data: Random Coefficients, Statistical Interactions, and Measurement Error

We consider two-level models where a continuous response R and continuous covariates C are assumed missing at random. Inferences based on maximum likelihood or Bayes are routinely made by estimating their joint normal distribution from observed data and . However, if the model for R given C includes random coefficients, interactions, or polynomial terms, their joint distribution will be nonstandard. We propose a family of unique factorizations involving selected “provisionally known random effects” u such that is normally distributed and u is a low-dimensional normal random vector; we approximate via adaptive Gauss-Hermite quadrature. For polynomial models, the approximation is exact but, in any case, can be made as accurate as required given sufficient computation time. The model incorporates random effects as explanatory variables, reducing bias due to measurement error. By construction, our factorizations solve problems of compatibility among fully conditional distributions that have arisen in Bayesian imputation based on the Gibbs Sampler. We spell out general rules for selecting u, and show that our factorizations can support fully compatible Bayesian methods of imputation using the Gibbs Sampler. Supplementary materials for this article are available online.

Random Effects Multinomial Processing Tree Models: A Maximum Likelihood Approach

The present article proposes and evaluates marginal maximum likelihood (ML) estimation methods for hierarchical multinomial processing tree (MPT) models with random and fixed effects. We assume that an identifiable MPT model with S parameters holds for each participant. Of these S parameters, R parameters are assumed to vary randomly between participants, and the remaining S-R parameters are assumed to be fixed. We also propose an extended version of the model that includes effects of covariates on MPT model parameters. Because the likelihood functions of both versions of the model are too complex to be tractable, we propose three numerical methods to approximate the integrals that occur in the likelihood function, namely, the Laplace approximation (LA), adaptive Gauss–Hermite quadrature (AGHQ), and Quasi Monte Carlo (QMC) integration. We compare these three methods in a simulation study and show that AGHQ performs well in terms of both bias and coverage rate. QMC also performs well but the number of responses per participant must be sufficiently large. In contrast, LA fails quite often due to undefined standard errors. We also suggest ML-based methods to test the goodness of fit and to compare models taking model complexity into account. The article closes with an illustrative empirical application and an outlook on possible extensions and future applications of the proposed ML approach.

A Mixed-Effects Model in Which the Parameters of the Autocorrelated Error Structure Can Differ between Individuals

Research in psychology has seen a rapid increase in the usage of experience sampling methods and daily diary methods. The data that result from using these methods are typically analyzed with a mixed-effects or a multilevel model because it allows testing hypotheses about the time course of the longitudinally assessed variable or the influence of time-varying predictors in a simple way. Here, we describe an extension of this model that does not only allow to include random effects for the mean structure but also for the residual variance, for the parameter of an autoregressive process of order 1 and/or the parameter of a moving average process of order 1. After we have introduced this extension, we show how to estimate the parameters with maximum likelihood. Because the likelihood function contains complex integrals, we suggest using adaptive Gauss-Hermite quadrature and Quasi-Monte Carlo integration to approximate it. We illustrate the models using a real data example and also report the results of a small simulation study in which the two integral approximation methods are compared.

Fast estimation of multiple group generalized linear latent variable models for categorical observed variables

A computationally efficient method for marginal maximum likelihood estimation of multiple group generalized linear latent variable models for categorical data is introduced. The approach utilizes second-order Laplace approximations of the integrals in the likelihood function. It is demonstrated how second-order Laplace approximations can be utilized highly efficiently for generalized linear latent variable models by considering symmetries that exist for many types of model structures. In a simulation with binary observed variables and four correlated latent variables in four groups, the method has similar bias and mean squared error compared to adaptive Gauss-Hermite quadrature with five quadrature points while substantially improving computational efficiency. An empirical example from a large-scale educational assessment illustrates the accuracy and computational efficiency of the method when compared against adaptive Gauss-Hermite quadrature with three, five, and 13 quadrature points.

Stochastic Convergence Rates and Applications of Adaptive Quadrature in Bayesian Inference

We provide the first stochastic convergence rates for a family of adaptive quadrature rules used to normalize the posterior distribution in Bayesian models. Our results apply to the uniform relative error in the approximate posterior density, the coverage probabilities of approximate credible sets, and approximate moments and quantiles, therefore, guaranteeing fast asymptotic convergence of approximate summary statistics used in practice. The family of quadrature rules includes adaptive Gauss-Hermite quadrature, and we apply this rule in two challenging low-dimensional examples. Further, we demonstrate how adaptive quadrature can be used as a crucial component of a modern approximate Bayesian inference procedure for high-dimensional additive models. The method is implemented and made publicly available in the aghq package for the R language, available on CRAN. Supplementary materials for this article are available online.

Using the Lambert Function to Estimate Shared Frailty Models with a Normally Distributed Random Intercept

Shared frailty models, that is, hazard regression models for censored data including random effects acting multiplicatively on the hazard, are commonly used to analyze time-to-event data possessing a hierarchical structure. When the random effects are assumed to be normally distributed, the cluster-specific marginal likelihood has no closed-form expression. A powerful method for approximating such integrals is the adaptive Gauss-Hermite quadrature (AGHQ). However, this method requires the estimation of the mode of the integrand in the expression defining the cluster-specific marginal likelihood: it is generally obtained through a nested optimization at the cluster level for each evaluation of the likelihood function. In this work, we show that in the case of a parametric shared frailty model including a normal random intercept, the cluster-specific modes can be determined analytically by using the principal branch of the Lambert function, . Besides removing the need for the nested optimization procedure, it provides closed-form formulas for the gradient and Hessian of the approximated likelihood making its maximization by Newton-type algorithms convenient and efficient. The Lambert-based AGHQ (LAGHQ) might be applied to other problems involving similar integrals, such as the normally distributed random intercept Poisson model and the computation of probabilities from a Poisson lognormal distribution.

On the Estimation of POD and LOD of Qualitative Microbiological Assays from a Multi-Laboratory Validation Study.

Jarvis et al. in 2019 (J. AOAC Int. 102: 1617-1623) estimated the mean laboratory effect (µ), standard deviation of laboratory effects (σ), probability of detection (POD), and level of detection (LOD) from a multi-laboratory validation study of qualitative microbiological assays using a random intercept complementary log-log model. Their approach estimated σ based on a Laplace approximation to the likelihood function of the model, but estimated µ from a fixed effectmodel due to a limitation in the MS Excel spreadsheet which was used by the authors to develop a calculation tool. We compared the estimates of µ and σ from three approaches (the Laplace approximation that estimates µ and σ simultaneously from the random intercept model, adaptive Gauss-Hermite quadrature (AGHQ), and the method of Jarvis et al.) and introduced an R Shiny app to implement the AGHQ using the widely used "lme4" R package. We conducted a simulation study to compare the accuracy of the estimates of µ and σ from the three approaches and compared the estimates of µ, σ, LOD, etc. between the R Shiny app and the spreadsheet calculation tool developed by Jarvis et al. for an example dataset. Our simulation study shows that, while the three approaches produce similar estimates of σ, the AGHQ has generally the best performance for estimating µ (and hence mean POD and LOD). The differences in the estimates between the R Shiny app and the spreadsheet were demonstrated using the example dataset. The AGHQ is the best method for estimating µ, POD, and LOD. The user-friendly R Shiny app provides a better alternative to the spreadsheet.

Mexhaz: An R Package for Fitting Flexible Hazard-Based Regression Models for Overall and Excess Mortality with a Random Effect

We present mexhaz, an R package for fitting flexible hazard-based regression models with the possibility to add time-dependent effects of covariates and to account for a twolevel hierarchical structure in the data through the inclusion of a normally distributed random intercept (i.e., a log-normally distributed shared frailty). Moreover, mexhazbased models can be fitted within the excess hazard setting by allowing the specification of an expected hazard in the model. These models are of common use in the context of the analysis of population-based cancer registry data. Follow-up time can be entered in the right-censored or counting process input style, the latter allowing models with delayed entries. The logarithm of the baseline hazard can be flexibly modeled with B-splines or restricted cubic splines of time. Parameters estimation is based on likelihood maximization: in deriving the contribution of each observation to the cluster-specific conditional likelihood, Gauss-Legendre quadrature is used to calculate the cumulative hazard; the cluster-specific marginal likelihoods are then obtained by integrating over the random effects distribution, using adaptive Gauss-Hermite quadrature. Functions to compute and plot the predicted (excess) hazard and (net) survival (possibly with cluster-specific predictions in the case of random effect models) are provided. We illustrate the use of the different options of the mexhaz package and compare the results obtained with those of other available R packages.

Multiple Imputation of Binary Multilevel Missing not at Random Data

SummaryWe introduce a selection model-based multilevel imputation approach to be used within the fully conditional specification framework for multiple imputation. Concretely, we apply a censored bivariate probit model to describe binary variables assumed to be missing not at random. The first equation of the model defines the regression model for the missing data mechanism. The second equation specifies the regression model of the variable to be imputed. The non-random selection of the binary data is mapped by correlations between the error terms of the two regression models. Hierarchical data structures are modelled by random intercepts in both equations. To fit the novel imputation model we use maximum likelihood and adaptive Gauss–Hermite quadrature. A comprehensive simulation study shows the overall performance of the approach. We test its usefulness for empirical research by applying it to a common problem in social scientific research: the emergence of educational aspirations. Our software is designed to be used in the R package mice.

A note on the accuracy of adaptive Gauss–Hermite quadrature

Summary Numerical quadrature methods are needed for many models in order to approximate integrals in the likelihood function. In this note, we correct the error rate given by Liu & Pierce (1994) for integrals approximated with adaptive Gauss–Hermite quadrature and show that the approximation is less accurate than previously thought. We discuss the relationship between the error rates of adaptive Gauss–Hermite quadrature and Laplace approximation, and provide a theoretical explanation of simulation results obtained in previous studies regarding the accuracy of adaptive Gauss–Hermite quadrature.

Effect of misspecification of random effects distribution on the performance of parameters estimation methods in binary logistic mixed models

Nous avons compare, a l’aide de la simulation, une methode d’estimation Bayesienne (INLA) a trois methodes classiques d’estimation (PQL, HLM et AGHQ) sous six distributions des effets aleatoires dans le cadre des modeles logistiques binaires mixtes. Les resultats ont montre la superiorite de AGHQ et HLM sur les autres pour toutes les distributions considerees dans le cas des effets fixes. Pour les effets aleatoires, les methodes classiques ont montre les meilleures performances pour les distributions symetriques consideres. AGHQ, HLM et INLA ont donne de meilleurs resultats par rapport a PQL pour les distributions normale et uniforme pour tous les echantillons consideres.

A Variational Maximization-Maximization Algorithm for Generalized Linear Mixed Models with Crossed Random Effects.

We present a variational maximization-maximization algorithm for approximate maximum likelihood estimation of generalized linear mixed models with crossed random effects (e.g., item response models with random items, random raters, or random occasion-specific effects). The method is based on a factorized variational approximation of the latent variable distribution given observed variables, which creates a lower bound of the log marginal likelihood. The lower bound is maximized with respect to the factorized distributions as well as model parameters. With the proposed algorithm, a high-dimensional intractable integration is translated into a two-dimensional integration problem. We incorporate an adaptive Gauss-Hermite quadrature method in conjunction with the variational method in order to increase computational efficiency. Numerical studies show that under the small sample size conditions that are considered the proposed algorithm outperforms the Laplace approximation.

Joint modelling of longitudinal and survival data: incorporating delayed entry and an assessment of model misspecification.

A now common goal in medical research is to investigate the inter‐relationships between a repeatedly measured biomarker, measured with error, and the time to an event of interest. This form of question can be tackled with a joint longitudinal‐survival model, with the most common approach combining a longitudinal mixed effects model with a proportional hazards survival model, where the models are linked through shared random effects. In this article, we look at incorporating delayed entry (left truncation), which has received relatively little attention. The extension to delayed entry requires a second set of numerical integration, beyond that required in a standard joint model. We therefore implement two sets of fully adaptive Gauss–Hermite quadrature with nested Gauss–Kronrod quadrature (to allow time‐dependent association structures), conducted simultaneously, to evaluate the likelihood. We evaluate fully adaptive quadrature compared with previously proposed non‐adaptive quadrature through a simulation study, showing substantial improvements, both in terms of minimising bias and reducing computation time. We further investigate, through simulation, the consequences of misspecifying the longitudinal trajectory and its impact on estimates of association. Our scenarios showed the current value association structure to be very robust, compared with the rate of change that we found to be highly sensitive showing that assuming a simpler trend when the truth is more complex can lead to substantial bias. With emphasis on flexible parametric approaches, we generalise previous models by proposing the use of polynomials or splines to capture the longitudinal trend and restricted cubic splines to model the baseline log hazard function. The methods are illustrated on a dataset of breast cancer patients, modelling mammographic density jointly with survival, where we show how to incorporate density measurements prior to the at‐risk period, to make use of all the available information. User‐friendly Stata software is provided. © 2015 The Authors. Statistics in Medicine Published by John Wiley & Sons Ltd.

Evaluation of a Bayesian Approach to Estimating Nonlinear Mixed-Effects Mixture Models

The growth mixture model has become increasingly popular, given the willingness to acknowledge developmental heterogeneity in populations. Typically, linear growth mixture models, based on polynomials or piecewise functions, are used in substantive applications and evaluated quantitatively through simulation. Growth mixture models that follow inherently nonlinear trajectories, referred to as nonlinear mixed-effects mixture models, have received comparatively little attention—likely due to estimation complexity. Previous work on the estimation of these models has involved multistep routines (Kelley, 2008), maximum likelihood estimation (MLE) via the E-M algorithm (Harring, 2005, 2012), Taylor series expansion and MLE within the structural equation modeling framework (Grimm, Ram, & Estabrook, 2010), and MLE by adaptive Gauss–Hermite quadrature (Codd & Cudeck, 2014). This article proposes and evaluates the use of Bayesian estimation with OpenBUGS (Lunn, Spiegelhalter, Thomas, & Best, 2009), a free program, and compares its performance with the Taylor series expansion approach. Finally, these estimation routines are used to evaluate the need for multiple latent classes to account for between-child differences in the development of reading ability.

Asymptotic properties of adaptive maximum likelihood estimators in latent variable models

Latent variable models have been widely applied in different fields of research in which the constructs of interest are not directly observable, so that one or more latent variables are required to reduce the complexity of the data. In these cases, problems related to the integration of the likelihood function of the model arise since analytical solutions do not exist. In the recent literature, a numerical technique that has been extensively applied to estimate latent variable models is the adaptive Gauss-Hermite quadrature. It provides a good approximation of the integral, and it is more feasible than classical numerical techniques in presence of many latent variables and/or random effects. In this paper, we formally investigate the properties of maximum likelihood estimators based on adaptive quadratures used to perform inference in generalized linear latent variable models.

Multidimensional Item Response Model for Nominal Variables

This article describes a multidimensional generalization of the nominal categories model that serves to estimate factorial models from nominal and ordinal observed responses, and includes a structural model for latent variables that distinguishes between endogenous and exogenous factors. The model includes a scale parameter for each response category in each factor. Item parameters relate the logit between categories to the vector of latent variables. The inferential framework is marginal maximum likelihood, implemented via static and adaptive Gauss–Hermite quadrature and Monte Carlo EM. The properties of estimators are investigated in a simulation study. An example with real data illustrates the utility of the model in analyzing local dependencies in testlets composed of multiple-choice items that are clustered in several groups around a common theme.

Multilevel mixed effects parametric survival models using adaptive Gauss-Hermite quadrature with application to recurrent events and individual participant data meta-analysis.

Multilevel mixed effects survival models are used in the analysis of clustered survival data, such as repeated events, multicenter clinical trials, and individual participant data (IPD) meta-analyses, to investigate heterogeneity in baseline risk and covariate effects. In this paper, we extend parametric frailty models including the exponential, Weibull and Gompertz proportional hazards (PH) models and the log logistic, log normal, and generalized gamma accelerated failure time models to allow any number of normally distributed random effects. Furthermore, we extend the flexible parametric survival model of Royston and Parmar, modeled on the log-cumulative hazard scale using restricted cubic splines, to include random effects while also allowing for non-PH (time-dependent effects). Maximum likelihood is used to estimate the models utilizing adaptive or nonadaptive Gauss-Hermite quadrature. The methods are evaluated through simulation studies representing clinically plausible scenarios of a multicenter trial and IPD meta-analysis, showing good performance of the estimation method. The flexible parametric mixed effects model is illustrated using a dataset of patients with kidney disease and repeated times to infection and an IPD meta-analysis of prognostic factor studies in patients with breast cancer. User-friendly Stata software is provided to implement the methods.